Kaplansky Theorem for completely regular spaces
Kaplansky Theorem for completely regular spaces
复制标题
DOI：
10.1090/s0002-9939-2014-11889-2
复制
发表时间：
2014-01
期刊：
影响因子：
--
通讯作者：
Lei Li;N. Wong
中科院分区：
文献类型：
--
作者：
Lei Li;N. Wong
设X，Y是由G-δ点组成的实紧空间或完全正则空间。设φ是来自C(X)的线性双射映射。C(X))到C(Y)(分别C(Y))。证明了如果φ保持f(X)6=0，∀x∈X，⇐⇒φ(F)(Y)6=0，∀y∈Y，则φ是由同胚φ：YφX产生的加权复合算子◦τ(F)=τ(1)·f→。这一结果也适用于度量空间上的其他好函数空间，如一致或Lipschitz连续函数。
Let X, Y be realcompact spaces or completely regular spaces consisting of Gδ-points. Let φ be a linear bijective map from C(X) (resp. C(X)) onto C(Y ) (resp. C(Y )). We show that if φ preserves nonvanishing functions, that is, f(x) 6= 0,∀x ∈ X, ⇐⇒ φ(f)(y) 6= 0,∀ y ∈ Y, then φ is a weighted composition operator φ(f) = φ(1) · f ◦ τ, arising from a homeomorphism τ : Y → X. This result is applied also to other nice function spaces, e.g., uniformly or Lipschitz continuous functions on metric spaces.